We consider the stochastic heat equation with a multiplicative colored noise term on ℝ2 d ≥ 1 in dimensions greater or equal to one. First, we prove convergence of a branching particle system in a random environment, to the stochastic heat equation with a linear noise term. For this stochastic partial differential equation with more general non-Lipschitz noise coefficients we show convergence of associated lattice systems, which are infinite dimensional stochastic differential equations with correlated noise terms, provided that uniqueness of the limit is known. In the course of the proof, we establish existence and uniqueness of solutions to the lattice systems, as well as a new existence result for solutions to the stochastic heat equation. The latter are shown to be jointly continuous in time and space under some mild additional assumptions.